Structure of Thin Irreducible Modules of a Q-polynomial Distance-Regular Graph

Let Gamma be a Q-polynomial distance-regular graph with vertex set X, diameter D geq 3 and adjacency matrix A. Fix x in X and let A*=A*(x) be the corresponding dual adjacency matrix. Recall that the Terwilliger algebra T=T(x) is the subalgebra of Mat_X(C) generated by A and A*. Let W denote a thin irreducible T-module. It is known that the action of A and A* on W induces a linear algebraic object known as a Leonard pair. Over the past decade, many results have been obtained concerning Leonard pairs. In this paper, these results will be applied to obtain a detailed description of W. In particular, we give a description of W in terms of its intersection numbers, dual intersection numbers and parameter array. Finally, we apply our results to the case in which Gamma has q-Racah type or classical parameters.